How to teach greatest common factor and lowest common multiple

Grade 4 to Grade 6

The greatest common factor (GCF) of two numbers is the largest number that divides into both. The lowest common multiple (LCM) is the smallest number both divide into. For 12 and 18 the GCF is 6 and the LCM is 36. The two are opposite directions, factors run down and stop, multiples run up and never end, so students need factors and multiples secure first.

How to teach it

1. Keep the words straight: factors are what goes into a number, multiples are what a number goes into.
2. For the GCF, list the factors of each number, ring the ones in both, and take the largest.
3. For the LCM, list multiples of each number until one appears in both lists, and take the smallest.
4. Show a real use for each: GCF for simplifying fractions and sharing into equal groups, LCM for adding fractions and repeating events lining up.
5. Introduce prime factorisation later as a faster method once the listing method is understood.

Worked example

For 12 and 18:

   factors of 12: 1, 2, 3, 4, 6, 12
   factors of 18: 1, 2, 3, 6, 9, 18
   common: 1, 2, 3, 6  -> GCF = 6

   multiples of 12: 12, 24, 36, 48
   multiples of 18: 18, 36, 54
   first shared -> LCM = 36

Common mistakes

• Swapping the two, giving a multiple when the greatest common factor is wanted.
• Taking a common factor that is not the greatest (choosing 3 instead of 6 for 12 and 18).
• Stopping the multiples lists too early and missing the first shared value.
• Thinking the LCM is always the two numbers multiplied together (it is only when they share no factors).